| L(s) = 1 | − 0.724·2-s − 1.47·4-s + 2.15·5-s − 0.509·7-s + 2.51·8-s − 1.56·10-s − 11-s − 4.17·13-s + 0.368·14-s + 1.12·16-s + 1.93·17-s − 6.89·19-s − 3.18·20-s + 0.724·22-s + 1.75·23-s − 0.340·25-s + 3.01·26-s + 0.751·28-s + 7.66·29-s − 0.682·31-s − 5.85·32-s − 1.40·34-s − 1.09·35-s + 0.848·37-s + 4.99·38-s + 5.43·40-s + 8.95·41-s + ⋯ |
| L(s) = 1 | − 0.512·2-s − 0.737·4-s + 0.965·5-s − 0.192·7-s + 0.889·8-s − 0.494·10-s − 0.301·11-s − 1.15·13-s + 0.0985·14-s + 0.282·16-s + 0.469·17-s − 1.58·19-s − 0.712·20-s + 0.154·22-s + 0.365·23-s − 0.0681·25-s + 0.592·26-s + 0.142·28-s + 1.42·29-s − 0.122·31-s − 1.03·32-s − 0.240·34-s − 0.185·35-s + 0.139·37-s + 0.809·38-s + 0.858·40-s + 1.39·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8019 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8019 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 0.724T + 2T^{2} \) |
| 5 | \( 1 - 2.15T + 5T^{2} \) |
| 7 | \( 1 + 0.509T + 7T^{2} \) |
| 13 | \( 1 + 4.17T + 13T^{2} \) |
| 17 | \( 1 - 1.93T + 17T^{2} \) |
| 19 | \( 1 + 6.89T + 19T^{2} \) |
| 23 | \( 1 - 1.75T + 23T^{2} \) |
| 29 | \( 1 - 7.66T + 29T^{2} \) |
| 31 | \( 1 + 0.682T + 31T^{2} \) |
| 37 | \( 1 - 0.848T + 37T^{2} \) |
| 41 | \( 1 - 8.95T + 41T^{2} \) |
| 43 | \( 1 - 11.9T + 43T^{2} \) |
| 47 | \( 1 - 0.714T + 47T^{2} \) |
| 53 | \( 1 - 6.72T + 53T^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 61 | \( 1 + 0.424T + 61T^{2} \) |
| 67 | \( 1 + 11.2T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 + 9.00T + 73T^{2} \) |
| 79 | \( 1 - 8.25T + 79T^{2} \) |
| 83 | \( 1 - 2.56T + 83T^{2} \) |
| 89 | \( 1 + 6.86T + 89T^{2} \) |
| 97 | \( 1 + 16.7T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.69069102265413208699517361863, −6.83339673400516553043036205694, −6.06070656502115465434221760489, −5.43235254608456188102394494059, −4.63459772782196708707243846899, −4.13376665603029552984650010908, −2.84672277362696331917371209639, −2.20622818129080203357140052557, −1.13698475028123408434504470468, 0,
1.13698475028123408434504470468, 2.20622818129080203357140052557, 2.84672277362696331917371209639, 4.13376665603029552984650010908, 4.63459772782196708707243846899, 5.43235254608456188102394494059, 6.06070656502115465434221760489, 6.83339673400516553043036205694, 7.69069102265413208699517361863
